A new complemented subspace for the Lorentz sequence spaces, with an application to its lattice of closed ideals

TL;DR

This paper constructs a new complemented subspace within Lorentz sequence spaces, revealing novel structural insights and expanding the known lattice of closed ideals in the space of bounded operators.

Contribution

It introduces a new 1-complemented subspace in Lorentz sequence spaces, distinct from classical subspaces, and applies this to identify a sixth element in the lattice of closed ideals.

Findings

Existence of a new complemented subspace in Lorentz spaces.

Explicit construction of the subspace for specific weights.

Identification of a new element in the lattice of closed ideals.

Abstract

We show that every Lorentz sequence space $d(\textbf{w},p)$ admits a 1-complemented subspace $Y$ distinct from $\ell_p$ and containing no isomorph of $d(\textbf{w},p)$. In the general case, this is only the second nontrivial complemented subspace in $d(\textbf{w},p)$ yet known. We also give an explicit representation of $Y$ in the special case $\textbf{w}=(n^{-\theta})_{n=1}^\infty$ ($0<\theta<1$) as the $\ell_p$-sum of finite-dimensional copies of $d(\textbf{w},p)$. As an application, we find a sixth distinct element in the lattice of closed ideals of $\mathcal{L}(d(\textbf{w},p))$, of which only five were previously known in the general case.